Improved Lower Bounds on the Classical Ramsey Numbers R(4,22) and R(4,25)

TL;DR

This paper advances the understanding of classical Ramsey numbers by constructing circulant graphs of prime order using higher-order residues, leading to improved lower bounds on R(4,22) and R(4,25).

Contribution

It introduces new constructions of circulant graphs based on quartic and higher-order residues to establish stronger lower bounds on specific Ramsey numbers.

Findings

New circulant graphs improve lower bounds on R(4,22) and R(4,25)

Use of quartic and higher-order residues in graph construction

Enhanced methods for establishing Ramsey number bounds

Abstract

Circulant graphs have been used to effectively establish lower bounds on many classical Ramsey numbers. Here, we construct circulant graphs of prime order that sharpen the best published lower bounds on two Ramsey numbers. Generalizing previous work in which quadratic and cubic residues were used to construct circulant graphs for the same purpose, we report on the use of quartic and higher-order residues.